An asymptotic expansion of the Kontorovich–Lebedev transform of damped oscillatory functions

An asymptotic expansion valid for large positive values of s is constructed for the integral transformF(s)=∫0∞ Kis(x)f(x)dxx,where Kis(x) denotes the modified Bessel function of the third kind of purely imaginary order. The expansion applies to functions f(x) that are analytic in the sector |arg(x)|⩽π/4 and that are exponentially damped and oscillatory as x→∞ in this sector.